Groundwater Modeling

34.9 Groundwater Modeling

The management of groundwater requires the capability of predicting subsurface flow and transport of solutes either under natural conditions or in response to human activities. These models are based on The management of groundwater requires the capability of predicting subsurface flow and transport of solutes either under natural conditions or in response to human activities. These models are based on

The continuity equation is (De Smedt, 1999)

rqab Í ( + ) + ˙ = -— ◊ q

∂ p ∂q ˘

() r

where q = the water content

a = the elastic compressibility coefficient of the porous formation

b = the compressibility coefficient of the water, both with dimensions [L 2 /F] p = the groundwater pressure q = the Darcian flux vector, namely the volumetric discharge which has components in the

three directions x, y, z r = the water density — = (∂/∂x,∂/∂y,∂/∂z) is the del operator and the dot represents the scalar product

The saturated groundwater flow equation is (De Smedt, 1999)

ËÁ o ËÁ ¯˜ ∂ z ËÁ ∂ z ¯˜ ∂ t

¯˜ h y ∂ y

where K h and K v = the horizontal and vertical hydraulic conductivities

h = the hydraulic head S o = the specific storage coefficient, namely the volume of water released per unit bulk volume of the saturated porous medium and per unit decline of the piezometric surface

If the flow is unsteady there is an initial condition at time t = 0 of the form h(x,y,z,0) = h o (x,y,z). There are three types of boundary conditions. In the first type the value of h is known at the boundary h(x b , y b ,z b ,t) = h b (t) where the subscript b refers to the flow boundary. The second type is a flux boundary condition when the amount of groundwater exchange at the boundary is known. It is of the form q b (x b ,y b ,z b ,t) = q b (t). The third type is a mixture of the two pervious types. Generally these partial differential equations are solved numerically along with the boundary condi- tions specific to the problem. Both finite difference and finite element methods can be used. The solute transport equation is (Konikow and Reilly, 1999) is

∂ () nC e ∂ Ê

( n Cv e i ) -¢ CW * - ∂ r t ∂ x

∂ t where

C = the solute concentration n e = the effective porosity of the porous medium

D ij = the coefficient of hydrodynamic dispersion (a second order tensor)

v i = the seepage velocity (q i /n e )

C ¢ = the concentration of the solute in the source or sink of fluid W* = the volumetric flux (positive for outflow, negative for inflow) —

C = the concentration of the species adsorbed on the solid (mass of solute/mass of solid) r b = the bulk density of the porous medium.

This equation is written for the case of linear equilibrium controlled sorption or ion-exchange reac- tions. The first term on the right hand side represents the change in concentration due to hydrodynamic dispersion. The last term changes in case of chemical rate control reactions or decay.

The complete solute transport model requires at least two equations: one equation for the flow and one for the solute transport. The velocities are obtained from the flow equation. For advectively domi- nated transport problems, the equations are hyperbolic partial differential equations. In this case the method of characteristics can be used.

For a more complete treatment of groundwater modeling, see for example, Konikow and Reilly (1999).